Question

A base of a triangle is along the line \(\mathrm{x}=\mathrm{b}\) and its length is 2b. If the area of triangle is \(b^{2}\), then the vertex of a triangle lies on the line \(\ldots \ldots\) (a) \(x=-b\) (b) \(x=0\) (c) \(\mathrm{x}=(\mathrm{b} / 2)\) (d) \(x=b\)

Short answer

The vertex of the triangle lies on the line \(x = b\). So, the correct option is (d) x = b.

Step-by-step solution

Recall the area formula for a triangle

The formula for calculating the area of a triangle is: Area = \(\frac{1}{2} \times\) base \(\times\) height

Write down the given information for the triangle

The given information for the triangle is: - Base is along the line x = b and has a length of 2b - Area of triangle is \(b^2\)

Substitute the given area and base into the area formula

Using the given information, substitute the area and base into the area formula: \(b^2\) = \(\frac{1}{2} \times\) (2b) \(\times\) height

Solve for the height of the triangle

Now, we can solve the equation for the height of the triangle: \(b^2\) = (b) \(\times\) height So, height = \(\frac{b^2}{b}\) = b

Determine the line on which the vertex of the triangle lies

The height of the triangle is also b. As the base of the triangle lies along the line x = b and has a length of 2b, the vertex of the triangle forms a right angle to the base. Hence, the vertex of the triangle must lie directly above the midpoint of the base. The midpoint of the base is \(\frac{1}{2} \times (2b)\) = b. Since the height of the triangle is equal to the base's midpoint, the vertex of the triangle lies on the line x = b. So, the correct option is (d) x = b.

Key concepts

Triangular Area Formula

The area of a triangle is a fundamental concept in geometry, crucial for understanding various properties of triangles and their applications. The most commonly used formula for calculating the area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \). This formula tells us that the area can be found by multiplying the length of the base by the height of the triangle and then dividing by two.

It's important to remember that the 'height' is the perpendicular distance from the base to the opposite vertex of the triangle. This formula works for all types of triangles, provided you have the necessary measurements. In practical situations, such as the given exercise, where a base and an area are specified, this formula can be rearranged to solve for missing variables, such as the height, which is a pivotal step for solving many geometry problems.

Coordinate Geometry

Coordinate geometry is an area of mathematics that allows us to analyze geometrical shapes using a system of coordinates. This system is typically two-dimensional, with horizontal and vertical axes referred to as the x-axis and y-axis, respectively. In the given exercise, the base of the triangle lies along a line which is defined by an equation (x = b), indicating it is parallel to the y-axis and has a fixed x-coordinate.

Understanding the relationship between the algebraic representation of shapes and their geometric properties is vital. For instance, the length, slope, and midpoint of the base line segment can be determined using principles of coordinate geometry, and from there, one can deduce the position of the triangle's vertex. This interplay between algebra and geometry helps to create a clear visual representation of abstract problems.

Solving for Height

Solving for the height in triangle problems can be approached startegically using the area formula. When the base and area are known, we can rearrange the area formula to find the height. For the given exercise, the calculation was made by setting the known area \( b^2 \) equal to the product of one half the base length and the height.

This rearrangement revealed the height to be equal to the base, which is a unique characteristic for an isosceles or an equilateral triangle when the base and height coincide. Grasping this concept not only helps in solving routine problems but also strengthens understanding of the triangle's properties, which can be utilized in complex geometric proof problems.

Properties of Triangles

Triangles are the simplest polygon and have fascinating properties that have vital applications in various fields. Some key properties include the sum of angles in a triangle always being 180 degrees, and the value of an exterior angle being equal to the sum of the opposite interior angles. In addition, the types of triangles (isosceles, equilateral, and right triangles) have their unique properties.

Equilateral triangles have all sides and angles equal, isosceles have at least two equal sides and angles, and right triangles have one 90-degree angle with the properties defined by the Pythagorean theorem. In the given solution, noting that the height is equivalent to the base indicates a special relationship between the sides, potentially suggesting an isosceles or equilateral property, barring additional context.

Midpoint Formula

The midpoint formula is a cornerstone of coordinate geometry. It provides a method for finding the exact center point of a line segment defined by its endpoints in a coordinate plane. The formula is given by \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \) where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the endpoints.

For example, if we want to find the midpoint of a horizontal base of a triangle with endpoints at (0,b) and (2b,b), we would apply the formula and get (b,b) as the midpoint. This ties into the current exercise where knowing the midpoint helps in determining the location of the vertex when the height and the area of the triangle are known. Hence, understanding the midpoint formula can be critical in solving a variety of problems involving triangles, especially when dealing with their representation on the Cartesian plane.